Merge branch 'chapter/Chapter4', key relationship diagram into develop

ch04.asciidoc

@@ -27,7 +27,12 @@ In most implementations, the private and public keys are stored together as a _k
 
 === Keys
 
-Your bitcoin wallet contains a collection of key pairs, each consisting of a private key and a public key. 
+Your bitcoin wallet contains a collection of key pairs, each consisting of a private key and a public key. The private key (k) is a number, usually picked at random. From the private key, we use elliptic curve multiplication, a one-way cryptographic function, to generate a public key (K). From the public key (K), we use a one-way cryptographic hash function to generate a bitcoin address (A). In this section we will start with generating the private key, look at the elliptic curve math that is used to turn that into a public key and finally, generate a bitcoin address from the public key. The relationship between private key, public key and bitcoin address is shown below:
+
+[[k_to_K_to_A]]
+.Private Key, Public Key and Bitcoin Address
+image::images/privk_to_pubK_to_addressA.png["privk_to_pubK_to_addressA"]
+
 
 ==== Private Keys
 
@@ -122,7 +127,7 @@ Starting with a private key in the form of a randomly generated number +k+, we m
 [[key_derivation]]
 where +k+ is the private key, +G+ is a fixed point on the curve called the _generator point_, ((("generator point"))) and +K+ is the resulting public key, another point on the curve. Since the generator point is always the same, a private key k multiplied with G will always produce the same public key K. 
 
-To visualize multiplication of a point with an integer, we will use the simpler elliptic curve over the real numbers - remember, the math is the same. Starting with the generator point G, we take the tangent of the curve at G until it crosses the curve again at another point. This new point is the negative of G+G, or -2G. Reflectign that point across the x-axis gives us 2G. If we take the tangent at 2G, it crosses the curve at -3G, which we can reflect on the x-axis to find 3G. Continuing this process, we can bounce around the curve finding the multiples of G, 2G, 3G, 4G etc. As you can see, a randomly selected large number k, when multiplied against the generator point G is like bouncing around the curve until we land on the point kG which is the public key. This process is irreversible, meaning that it is infeasible to find the factor k (the secret k) in any way other than trying all multiples of G (1G, 2G, 3G etc) in a brute-force search for k. Since k can be an enormous number, that brute-force search would take forever. 
+To visualize multiplication of a point with an integer, we will use the simpler elliptic curve over the real numbers - remember, the math is the same. Starting with the generator point G, we take the tangent of the curve at G until it crosses the curve again at another point. This new point is the negative of G+G, or -2G. Reflecting that point across the x-axis gives us 2G. If we take the tangent at 2G, it crosses the curve at -3G, which we can reflect on the x-axis to find 3G. Continuing this process, we can bounce around the curve finding the multiples of G, 2G, 3G, 4G etc. As you can see, a randomly selected large number k, when multiplied against the generator point G is like bouncing around the curve until we land on the point kG which is the public key. This process is irreversible, meaning that it is infeasible to find the factor k (the secret k) in any way other than trying all multiples of G (1G, 2G, 3G etc) in a brute-force search for k. Since k can be an enormous number, that brute-force search would take forever. 
 
 
 [[ecc_illustrated]]